A Globally Coupled Pressure Method for the Discretization of the Tensor-Pressure Equation on Non-K-orthogonal Grids

Abstract

Summary Complex permeability tensors together with general nonorthogonal and unstructured grids pose great challenges to reservoir simulation. The widely used two-point flux approximation (TPFA) is inadequate for a rigorous discretization of the flow equations on such challenging grids. Multipoint flux approximation (MPFA) methods have been proposed to meet the challenges and are currently being deployed in next-generation simulators. In this work, we propose an alternative flux-continuous cell-centered finite-volume method called the globally coupled pressure (GCP) method to discretize the pressure equation on general grids with full permeability tensors. To accurately construct fluxes through control-volume interfaces, pressure at the centroid of all interfaces is introduced as auxiliary unknowns. Flux continuity across each interface gives one equation. Assembling all the flux-continuity equations together gives a system of linear equations that can be solved simultaneously for all the auxiliary unknowns. Flux across control-volume interfaces can then be approximated with the pressure values at control-volume centers only. The fundamental difference between the GCP method and MPFA methods is that, in the latter, auxiliary unknowns are locally coupled within an interaction region and then eliminated in a local stencil by imposing flux-continuity conditions, whereas in the former, all the auxiliary unknowns are globally coupled and can only be eliminated in a global stencil. Consequently, control volumes in our GCP method are directly associated with the edges of the original grid and not by means of a dual grid overlaid and allied with the centers of the grid. Two variants of the GCP method are presented here, and extensive numerical experiments are conducted to test the performance of the GCP methods. The results show that both variants of our GCP method are in good agreement with the classical MPFA-O method on non-K-orthogonal grids for less-challenging problems. Convergence studies reveal that the first variant of our GCP method has slower convergence rates than the MPFA-O method for some problems. However, the second variant of GCP has comparable, and in some cases, better convergence properties compared with the MPFA-O method. With numerical experiments, we further investigate monotonicity properties of our GCP method on highly anisotropic media. For Dirichlet boundary conditions, our GCP methods also suffer from nonphysical oscillations, with some degrees of improvement over the MFPA-O method. When no-flow boundary conditions are used, our GCP method is much more robust and does not produce spurious boundary extrema as MPFA methods do. Finally, we extend our GCP method to fully unstructured grids, and the results show that it is also more robust than the MPFA-O method on unstructured grids.